The Fuzzy Assignment Problem (FAP) is a classic combinatorial optimization problem that has received a lot of attention. FAP has a wide range of uses. We suggest a new algorithm that combines to solve the FAP in this paper. Each column is maximized during the optimization process, and the best choice with the lowest cost is selected. The proposed method follows a standard methodology, is simple to execute, and takes less effort to compute. An order to obtain the best solution, the assignment problem is specifically solved here. We looked at how well trapezoidal fuzzy numbers performed. Then, to convert crisp numbers, we use the robust ranking method for trapezoidal fuzzy numbers. The optimality of the result provided by this new method is clarified by a numerical example.

1. https://iaeme.com/Home/journal/IJARET 390 editor@iaeme.com
International Journal of Advanced Research in Engineering and Technology (IJARET)
Volume 10, Issue 1, January-February 2019, pp.390-398, Article ID: IJARET_10_01_040
Available online at https://iaeme.com/Home/issue/IJARET?Volume=10&Issue=1
ISSN Print: 0976-6480 and ISSN Online: 0976-6499
DOI: https://doi.org/10.34218/IJARET.10.1.2019.040
© IAEME Publication Scopus Indexed
NEW APPROACH FOR SOLVING FUZZY
TRIANGULAR ASSIGNMENT BY ROW MINIMA
METHOD
V. Vanitha
Lecturer, Department of Mathematics, Tamilnadu Polytechnic College,
Madurai, Tamil Nadu, India
ABSTRACT
The Fuzzy Assignment Problem (FAP) is a classic combinatorial optimization
problem that has received a lot of attention. FAP has a wide range of uses. We suggest
a new algorithm that combines to solve the FAP in this paper. Each column is
maximized during the optimization process, and the best choice with the lowest cost is
selected. The proposed method follows a standard methodology, is simple to execute,
and takes less effort to compute. An order to obtain the best solution, the assignment
problem is specifically solved here. We looked at how well trapezoidal fuzzy numbers
performed. Then, to convert crisp numbers, we use the robust ranking method for
trapezoidal fuzzy numbers. The optimality of the result provided by this new method is
clarified by a numerical example.
Key words: Trapezoidal Fuzzy Number, Robust Ranking Method, Fuzzy Assignment
Problem, New Algorithm.
Cite this Article: V. Vanitha, New Approach for Solving Fuzzy Triangular Assignment
by Row Minima Method, International Journal of Advanced Research in Engineering
and Technology (IJARET), 10(1), 2019, pp. 390-398.
https://iaeme.com/Home/issue/IJARET?Volume=10&Issue=1
1. INTRODUCTION
The Assignment Problem (AP) may be a sort of applied mathematics problem (LPP) during
which the goal is to assign variety of sources to an equal number of destinations at the minimum
available cost (or maximum profit). The assignment problem may be a basic combinatorial
optimization problem within the branch of mathematics referred to as optimization and
operations analysis. Various methods for solving the assignment problem are introduced, and a
number of other papers on the topic are written. The mathematical formulation of the matter
proposes that new algorithms be built to seek out optimal Assignment Problem solutions.
Assume you've got an assignment problem with n machines and n jobs, and you would like to
scale back the general cost or time by assigning each machine to at least one and just one
employee.

2. New Approach for Solving Fuzzy Triangular Assignment by Row Minima Method
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The aim of the assignment problem is to allocate variety of sources to an equal number of
destinations at all-time low possible expense (or maximum profit) on a one-to-one basis. The
assignment problems are often seen during a number of situations. Allocation of n workers to
‘n’ machinery during a factory. To Resolve Assignment issues The Hungarian method is that
the commonest, but it appears to be repetitive as compared to the iterative method. The
algorithmic step involves finding the utmost (or minimum) element in each row and subtracting
each element of the row by using the maximum (or minimum) element so as to get any zeros
within the cost matrix, after which an entire assignment comes by allocating 0's based location.
In this article, we presented a method for solving the Fuzzy Assignment Problem (FAP)
with a fuzzy cost. The objective function is often treated as a fuzzy number, subject to these
sharp constraints, since the objectives are to reduce total cost or maximize total gain. Then,
using Robust's ranking method, rank the objective values of the objective function in order to
transform the Fuzzy Assignment Problem into a crisp one that can be solved using traditional
solution approaches. The aim is to transform a problem with fuzzy parameters into a problem
with crisp parameters, which can then be solved with a new algorithm process. Many ranking
methods, such as Robust's ranking system, which satisfies the properties of compensation,
linearity, and addictiveness, will explain the dominance of fuzzy numbers. The fundamental
concepts of fuzzy sets, fuzzy numbers, and fuzzy linear programming were also covered by
Srinivasan et al [7&8]. (FLP). A robusts ranking is discussed by Nagarajan et al. [2–5].
Thiruppathiet has proposed a new algorithm, al[11&3]. Srinivasanet. al[6] focuses on a single
task in order to find the best answer. The operation analysis has been discussed by Taha[9].
Thakreet.al[10] has discussed various methods for achieving the lowest rate. Zadeh[12] was the
first to propose the concept of fuzzy sets.
2. PRELIMINARIES
2.1. Definition: (Fuzzy Set)
A membership function mapping the components of a domain space or universe of discourse X
to the unit interval [0,1] defines a fuzzy set. ( ) ( )  
1
,
0
:
. →
X
x
e
i A



2.2. Definition: (Fuzzy number)
Assume A
 is a classical set, and ( )
x
A


 is a function from A
 to [0,1]. The membership function
( )
x
A


 of a fuzzy set A
 is known as ( )
( )
 A
x
;
, 



 
 x
x
A A
= and ( )  
0,1

 x
A



Definition: (Trapezoidal Fuzzy Number)
A trapezoidal fuzzy number A
 is represented by ( )
1
;
,
,
, d
c
b
a
A =
 with the membership
function
( )













−
−




−
−
=
otherwise
d
x
c
c
d
x
d
c
x
b
b
x
a
a
b
a
x
x
A
0
1




3. V. Vanitha
https://iaeme.com/Home/journal/IJARET 392 editor@iaeme.com
Figure 1 Trapezoidal Fuzzy Number
2.4. Magnitude Ranking Techniques
Magnitude ranking technique which satisfies compensation, linearity and additive properties
and provides results that is reliable with human perception. Given a complex fuzzy number 𝑎
̃,
the Magnitude Ranking method is defined by
( )
12
5
5
a
~ d
c
b
a
Mag
+
+
+
=
Magnitude ranking method ( )
a
M ~ gives the representative value of the fuzzy number ã . It
satisfies the linearity and additive properties.
3. MATHEMATICAL FORMULATION
3.1. The General Assignment Problem
Suppose there are ‘n’ people and ‘n’ jobs. Each job must be done by exactly one person; also
each person can do, at most, one job. The problem is to assign jobs to the people so as to
minimize the total cost of completing all of the jobs. The general assignment problem can be
mathematically stated as follows:
Minimize Z=
n
=
i
n
=
j
c
1 1
ij
ij x
Subject to
1, 1,2,....
1
n
x i n
ij
j
= =

=
1, 1,2,......
1
n
x j n
ij
i
= =

=
0,
0 or 1=
1,
th th
if i row isnot assigned to j column
x =
ij th th
if i row isnot assigned to j column






4. New Approach for Solving Fuzzy Triangular Assignment by Row Minima Method
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3.2. Fuzzy Assignment Problem
Minimize Z =
n
=
i
n
=
j
c
1 1
ij
ij x
~
Subject to
1, 1,2,....
1
n
x i n
ij
j
= =

=
1, 1,2,......
1
n
x j n
ij
i
= =

=
0,
0 or 1=
1,
th th
if i row isnot assigned to j column
x =
ij th th
if i row isnot assigned to j column





4. NEW ALGORITHMIC APPROACH FOR SOLVING FUZZY
ASSIGNMENT PROBLEM
Step 1 : Prepare the cost matrix. Consider the rows for people and the columns for work. Check
whether the given matrix is balanced or not. Suppose the matrix is not a square matrix, then
make it as a square matrix by adding fuzzy zero element rows or fuzzy zero element columns
accordingly.
Step 2 : Convert the fuzzy cost to crisp cost with the help of ranking function.
Step 3 : Find out the highest cost of the matrix in each column and write in the same column,
which is called as penalty.
Step 4 : Choose the minimum penalty and divide the each cost of the matrix.
Step 5 : Find out the minimum cost of the matrix and delete that row and column.
Step 6 : Repeat the steps 3 to 5.
Step 7 : Obtained the optimal cost.
5. NUMERICAL EXAMPLE
Let us consider a fuzzy assignment problem with rows representing four workers Job1, Job2,
Job3 and Job4 and columns representing four people A, B, C, D. The cost matrix Cij with
trapezoidal fuzzy numbers given in the following table
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3,5,6,7 5,8,11,12 9,10,11,15 5,8,10,11
7,8,10,11 3,5,6,7 6,8,10,12 5,8,9,10
2,4,5,6 5,7,10,11 8,11,13,15 4,6,7,10
6,8,10,12 2,5,6,7 5,7,10,11 2,4,5,7
 
 
 
 
 
 
 

5. V. Vanitha
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Step – 1
Table 1 Fuzzy assignment problem
I II III IV
A ( )
3,5,6,7 ( )
5,8,11,12 ( )
9,10,11,15 ( )
5,8,10,11
B ( )
7,8,10,11 ( )
3,5,6,7 ( )
6,8,10,12 ( )
5,8,9,10
C ( )
2,4,5,6 ( )
5,7,10,11 ( )
8,11,13,15 ( )
4,6,7,10
D ( )
6,8,10,12 ( )
2,5,6,7 ( )
5,7,10,11 ( )
2,4,5,7
Step – 2
Trapezoidal fuzzy cost matrix is balanced one. Fuzzy assignment problem can be formulated in
the following
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 44
43
42
41
34
33
32
31
24
23
22
21
14
13
12
11
2,4,5,7
M
5,7,10,11
M
2,5,6,7
M
6,8,10,12
M
4,6,7,10
M
8,11,13,15
M
5,7,10,11
M
2,4,5,6
M
5,8,9,10
M
6,8,10,12
M
3,5,6,7
M
7,8,10,11
M
5,8,10,11
M
9,10,11,15
M
5,8,11,12
M
3,5,6,7
M
Z
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Min
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
( )
12
5
5
a
~ d
c
b
a
Mag
+
+
+
=
( ) ( )
7
,
6
,
5
,
3
~
11 R
a
M =
( ) ( ) ( )
12
7
6
5
5
5
3
7
,
6
,
5
,
3
+
+
+
=
M
( ) 417
.
5
7
,
6
,
5
,
3 =
M
Similarly,
( ) 333
.
9
12
,
11
,
8
,
5 =
M
( ) 750
.
10
15
,
11
,
10
,
9 =
M
( ) 833
.
8
11
,
10
,
8
,
5 =
M
( ) 000
.
9
11
,
10
,
8
,
7 =
M
( ) 417
.
5
7
,
6
,
5
,
3 =
M
( ) 000
.
9
12
,
10
,
8
,
6 =
M
( ) 333
.
8
10
,
9
,
8
,
5 =
M
( ) 417
.
4
6
,
5
,
4
,
2 =
M
( ) 417
.
8
11
,
10
,
7
,
5 =
M
( ) 917
.
11
15
,
13
,
11
,
8 =
M

6. New Approach for Solving Fuzzy Triangular Assignment by Row Minima Method
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( ) 583
.
6
10
,
7
,
6
,
4 =
M
( ) 000
.
9
12
,
10
,
8
,
6 =
M
( ) 333
.
5
7
,
6
,
5
,
2 =
M
( ) 417
.
8
11
,
10
,
7
,
5 =
M
( ) 500
.
4
7
,
5
,
4
,
2 =
M
With the help of the above values, we can get the convenient assignment problem














500
.
4
417
.
8
333
.
5
000
.
9
583
.
6
917
.
11
417
.
8
417
.
4
333
.
8
000
.
9
417
.
5
000
.
9
833
.
8
750
.
10
333
.
9
417
.
5
Step – 3 Find out the highest cost of the matrix in each column and write in the same column,
which is called as penalty.
Table 2 Assignment problem
I II III IV
A 5.417 9.333 10.750 8.833
B 9.000 5.417 9.000 8.333
C 4.417 8.417 11.917 6.583
D 9.000 5.333 8.417 4.500
Penalty 9 9.333 11.917 8.833
Step - 4 Choose the minimum penalty and divide the each cost of the matrix.
Table 3 Iteration – 1
I II III IV
A 5.417/8.833 9.333/8.833 10.750/8.833 8.833/8.833
B 9.000/8.833 5.417/8.833 9.000/8.833 8.333/8.833
C 4.417/8.833 8.417/8.833 11.917/8.833 6.583/8.833
D 9.000/8.833 5.333/8.833 8.417/8.833 4.500/8.833
Step - 5 Find out the minimum cost of the matrix and delete that row and column.
Table 4 Iteration – 2
I II III IV
A 0.613 1.057 1.217 1
B 1.019 0.613 1.019 0.943
C 0.500 0.953 1.349 0.745
D 1.019 0.604 0.953 (0.509)
Table 5 Iteration – 3
I II III
A 0.613 1.057 1.217
B 1.019 0.613 1.019
C 0.500 0.953 1.349

7. V. Vanitha
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Job D assigns to Person IV.
Repeat the steps 3 to 5, until assigning each row and column.
Table 6 Iteration – 4
Table 7 Iteration – 5
Table 8 Iteration – 6
Job C Assign to Person I.
Table 9 Iteration – 7
Table 10 Iteration – 8
II III
A 9.333 10.750
B 5.417 9.000
Penalty (9.333) 10.750
Table 11 Iteration – 9
I II III
A 5.417 9.333 10.750
B 9.000 5.417 9.000
C 4.417 8.417 11.917
Penalty (9.000) 9.333 11.917
I II III
A 5.417/9 9.333/9 10.750/9
B 9.000/9 5.417/9 9.000/9
C 4.417/9 8.417/9 11.917/9
I II III
A 0.602 1.037 1.194
B 1 0.602 1
C (0.491) 0.935 1.324
II III
A 1.037 1.194
B 0.602 1
II III
A 9.333/9.333 10.750/9.333
B 5.417/9.333 9.000/9.333

8. New Approach for Solving Fuzzy Triangular Assignment by Row Minima Method
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Table 12 Iteration – 10
Job B Assign to Person II.
Job A assign to Person III
Table 13 Optimal Solution – 11
I II III IV
A 5.417 9.333 10.750 8.833
B 9.000 5.417 9.000 8.333
C 4.417 8.417 11.917 6.583
D 9.000 5.333 8.417 4.500
Optimum assignment schedule A→III, B→II, C→I, D→IV
Optimal (Minimum) assignment cost = 10.750 + 5.417 + 4.417 + 4.500 = 25.084
6. CONCLUSION
This paper introduced a new simple and effective method for solving the fuzzy assignment
problem. This method applies to all types of fuzzy assignment problems. The new approach is
a standardized technique that is simple to implement and can be used for any form of assignment
problem, whether the objective function is maximized or minimized.
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B (0.580) 0.964

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